Multiplicative Jensen's formula and quantitative global theory of one-frequency Schr\"odinger operators

TL;DR

This paper develops a multiplicative Jensen's formula for one-frequency Schrödinger operators using dual Lyapunov exponents, providing new insights and quantitative tools for spectral theory and physics applications.

Contribution

It introduces dual Lyapunov exponents and a multiplicative Jensen's formula, offering a new proof and quantitative analysis of Avila's global theory for Schrödinger cocycles.

Findings

Provides a new proof of Avila's global theory

Quantitatively explains critical regimes and acceleration

Enables powerful spectral and physics applications

Abstract

We introduce the concept of dual Lyapunov exponents, leading to a multiplicative version of the classical Jensen's formula for one-frequency analytic Schr\"odinger cocycles. This formula, in particular, gives a new proof and a quantitative version of the fundamentals of Avila's global theory \cite{avila}, fully explaining the behavior of complexified Lyapunov exponent through the dynamics of the dual cocycle. In particular, concepts of (sub/super) critical regimes and acceleration are all explained (in a quantitative way) through the duality approach. This leads to a number of powerful spectral and physics applications